Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 378376, 8 pages
http://dx.doi.org/10.1155/2013/378376
Research Article
Finite-Time Synchronization of Singular Hybrid
Coupled Networks
Cong Zheng1 and Jinde Cao1,2
1
Research Center for Complex Systems and Network Sciences, and Department of Mathematics, Southeast University,
Nanjing 210096, China
2
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Jinde Cao; jdcao@seu.edu.cn
Received 20 December 2012; Accepted 11 February 2013
Academic Editor: Jong Hae Kim
Copyright © 2013 C. Zheng and J. Cao. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper investigates finite-time synchronization of the singular hybrid coupled networks. The singular systems studied in this
paper are assumed to be regular and impulse-free. Some sufficient conditions are derived to ensure finite-time synchronization of
the singular hybrid coupled networks under a state feedback controller by using finite-time stability theory. A numerical example
is finally exploited to show the effectiveness of the obtained results.
1. Introduction
In recent years, singular systems, also known as descriptor systems, generalized state-space systems, differentialalgebraic systems, or semistate systems, are attracting more
and more attentions from many fields of scientific research
because they can better describe a larger class of dynamic
systems than the regular ones. Many results of regular systems
have been extended to the area about singular systems such
as [1–21]. For example, stability (robust stability or quadratic
stability) and stabilization for singular systems have been
studied via LMI approach in [2–8]; robust control (or 𝐻2 ,
𝐻∞ control and robust dissipative filtering) for singular
systems has been discussed in [9–16]; synchronization (or
state estimation) for singular complex networks has been
considered in [17–21].
Synchronization is an interesting and important characteristic in the coupled networks. There are a lot of results
in regular coupled networks. Recently, some authors study
synchronization of the singular systems such as [17–21] and
the references therein. In [17], Xiong et al. introduced the
singular hybrid coupled systems to describe complex network
with a special class of constrains. They gave a sufficient
condition for global synchronization of singular hybrid
coupled system with time-varying nonlinear perturbation
based on Lyapunov stability theory. Synchronization issues
are studied for singular systems with delays by using Linear
Matrix Inequality (LMI) approach [18]. Koo et al. considered
synchronization of singular complex dynamical network
with time-varying delays [19]. Li et al. in [20] investigated
synchronization and state estimation for singular complex
dynamical networks with time-varying delays. Li et al. in
[21] investigated robust 𝐻∞ control of synchronization for
uncertain singular complex delayed networks with stochastic
switched coupling.
Finite-time synchronization or finite-time control is
interesting topic for its practical application. There are some
results on finite-time stability [22–26], finite-time synchronization [27–33], finite-time consensus or agreement [34–
37], and finite-time observers [38]. However, these results are
obtained for regular systems. Up to now, to the best of our
knowledge, few authors studied finite-time synchronization
of singular hybrid coupled systems whose structures are more
complex than those in [27–33]. Considering the important
role of synchronization of complex networks, the finite-time
synchronization of singular hybrid coupled networks is worth
studying.
Motivated by the previous discussions, in this paper,
we investigate finite-time synchronization of singular hybrid
complex systems. Some sufficient conditions for it are
2
Journal of Applied Mathematics
obtained by the state feedback controller based on the finitetime stability theory. Finally, a numerical example is exploited
to illustrate the effectiveness of the obtained result.
The rest of this paper is organized as follows. In Section 2,
a singular hybrid coupled system is given, and some preliminaries are briefly outlined. In Section 3, some sufficient
criteria are derived for the finite-time synchronization of
the proposed singular system by the feedback controller. In
Section 4, an example is provided to show the effectiveness of
the obtained results. Some conclusions are finally drawn in
Section 5.
2. Model Formulation and Some Preliminaries
(1)
𝑖 = 1, 2, . . . , 𝑁,
where 𝑥𝑖 (𝑡) = (𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡), . . . , 𝑥𝑖𝑛 (𝑡))𝑇 ∈ R𝑛 represents the
state vector of the 𝑖th node, 𝐴, 𝐸 ∈ R𝑛×𝑛 are constant matrices,
and 𝐸 may be singular. Without loss of generality, we will
assume that 0 < rank(𝐸) = 𝑟 < 𝑛. 𝑓(𝑥𝑖 (𝑡), 𝑡) is a vector-value
function. The constant 𝑐 > 0 denotes the coupling strength,
and Γ = diag(𝛾1 , 𝛾2 , . . . , 𝛾𝑛 ) ∈ R𝑛×𝑛 is inner-coupling matrix
between nodes. 𝐵 = (𝑏𝑖𝑗 )𝑁×𝑁 describes the linear coupling
configuration of the network, which satisfies
𝑏𝑖𝑗 = 𝑏𝑗𝑖 ,
𝑁
𝑏𝑖𝑖 = − ∑ 𝑏𝑖𝑗 ,
for 𝑖 ≠ 𝑗,
𝑖 = 1, 2, . . . , 𝑁.
𝑁
+ 𝑐 ∑𝑏𝑖𝑗 (𝑡) Γ𝑒𝑗 (𝑡) ,
𝑖 = 1, 2, . . . , 𝑁.
(2)
𝑗=1,𝑗 ≠ 𝑖
Remark 1. If rank(𝐸) = 𝑛, then system (1) is a general nonsingular coupled network. We will also give a sufficient condition
of the finite-time synchronization for this circumstance. See
Corollary 10.
Definition 2. The singular system (1) is said to be synchronized in the finite time, if for a suitable designed feedback
controller, there exists a constant 𝑡∗ > 0 (which depends on
𝑇
the initial vector value 𝑥(0) = (𝑥1𝑇 (0), 𝑥2𝑇 (0), . . . , 𝑥𝑁
(0))𝑇 ),
such that lim𝑡 → 𝑡∗ ‖ 𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡) ‖= 0 and ‖ 𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡) ‖≡ 0
for 𝑡 > 𝑡∗ , 𝑖, 𝑗 = 1, 2, . . . , 𝑁.
Assumption 3. Assume that the singular system (1) is connected in the sense that there are no isolated clusters; that is,
the matrix 𝐵 is an irreducible matrix.
With Assumption 3, we obtain that zero is an eigenvalue
of 𝐵 with multiplicity 1, and all the other eigenvalues of
𝐵 are strictly negative, which are denoted by 0 = 𝜆 1 >
𝜆 2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆 𝑁. At the same time, since 𝐵 is a symmetric
matrix, there exists a unitary matrix 𝑊 = (𝑊1 , 𝑊2 , . . . , 𝑊𝑁) ∈
R𝑛×𝑛 such that 𝐵 = 𝑊Λ𝑊𝑇 with 𝑊𝑊𝑇 = 𝐼 and Λ =
diag(𝜆 1 , 𝜆 2 , . . . , 𝜆 𝑁).
(3)
𝑗=1
Let 𝑒(𝑡) = (𝑒1 (𝑡), 𝑒2 (𝑡), . . . , 𝑒𝑁(𝑡)), 𝑦(𝑡) = 𝑒(𝑡)𝑊; then
system (3) can be written as
𝐸𝑦̇ (𝑡) = 𝐴𝑦 (𝑡) + 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊 + 𝑐Γ𝑦 (𝑡) Λ,
𝑁
𝑗=1
𝐸𝑒𝑖̇ (𝑡) = 𝐴𝑒𝑖 (𝑡) + 𝑓 (𝑥𝑖 (𝑡) , 𝑡) − 𝑓 (𝑠 (𝑡) , 𝑡)
𝐸𝑒 ̇ (𝑡) = 𝐴𝑒 (𝑡) + 𝐹 (𝑒 (𝑡) , 𝑡) + 𝑐Γ𝑒 (𝑡) 𝐵𝑇 ,
Consider a singular hybrid coupled system as follows:
𝐸𝑥𝑖̇ (𝑡) = 𝐴𝑥𝑖 (𝑡) + 𝑓 (𝑥𝑖 (𝑡) , 𝑡) + 𝑐 ∑ 𝑏𝑖𝑗 (𝑡) Γ𝑥𝑗 (𝑡) ,
Let 𝑠(𝑡) be a function to which all 𝑥𝑖 (𝑡) are expected to
synchronize in the finite time. That is, the synchronization
̇ =
state is 𝑠(𝑡). Suppose that 𝑠(𝑡) satisfies the equation 𝐸𝑠(𝑡)
𝐴𝑠(𝑡) + 𝑓(𝑠(𝑡), 𝑡). Let 𝑒𝑖 (𝑡) = 𝑥𝑖 (𝑡) − 𝑠(𝑡), 𝑖 = 1, 2, . . . , 𝑁. We
can obtain the following singular error system:
(4)
where 𝐹(𝑒(𝑡), 𝑡) = (𝑓(𝑥1 (𝑡), 𝑡) − 𝑓(𝑠(𝑡), 𝑡), 𝑓(𝑥2 (𝑡), 𝑡) −
𝑓(𝑠(𝑡), 𝑡), . . . , 𝑓(𝑥𝑁(𝑡), 𝑡)−𝑓(𝑠(𝑡), 𝑡)), 𝑦(𝑡) = (𝑦1 (𝑡), 𝑦2 (𝑡), . . . ,
𝑦𝑁(𝑡)), and 𝑦𝑖 (𝑡) = 𝑒(𝑡)𝑊𝑖 ∈ R𝑛 , 𝑖 = 1, 2, . . . , 𝑁. Then, system
(4) can be written as
𝐸𝑦𝑖̇ (𝑡) = 𝐴𝑦𝑖 (𝑡) + 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 + 𝑐𝜆 𝑖 Γ𝑦𝑖 (𝑡)
= (𝐴 + 𝑐𝜆 𝑖 Γ) 𝑦𝑖 (𝑡) + 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 .
(5)
Therefore, the finite-time synchronization problem of
system (1) is equivalent to the finite-time stabilization of
system (5) at the origin under the suitable controllers 𝑢𝑖 ,
𝑖 = 1, 2, . . . , 𝑁.
Assumption 4. Assume that there exist nonnegative constants
𝐿 𝑖 such that
󵄩
󵄩
󵄩󵄩
󵄩
󵄩󵄩𝑓 (𝑥𝑖 (𝑡) , 𝑡) − 𝑓 (𝑠 (𝑡) , 𝑡)󵄩󵄩󵄩 ≤ 𝐿 𝑖 󵄩󵄩󵄩𝑥𝑖 (𝑡) − 𝑠 (𝑡)󵄩󵄩󵄩 ,
𝑖 = 1, 2, . . . , 𝑁.
(6)
Assumption 5. There exist matrices 𝑃𝑖 such that
𝐸𝑇 𝑃𝑖 = 𝑃𝑖𝑇 𝐸 ≥ 0,
𝑖 = 1, 2, . . . , 𝑁,
𝐴𝑇 𝑃1 + 𝑃1𝑇 𝐴 < 0,
(𝐴 + 𝑐𝜆 𝑖 Γ)𝑇 𝑃𝑖 + 𝑃𝑖𝑇 (𝐴 + 𝑐𝜆 𝑖 Γ) ≤ −𝜂𝑖 𝐼,
𝑖 = 2, . . . , 𝑁,
(7)
(8)
where 𝜂𝑖 > 2𝐿(𝑁 − 1) ‖ 𝑃𝑖 ‖, 𝐿 = ∑𝑁
𝑖=1 𝐿 𝑖 .
Lemma 6 (see [26]). Suppose that the function 𝑉(𝑡) : [𝑡0 ,
∞) → [0, ∞) is differentiable (the derivative of 𝑉(𝑡) at 𝑡0
̇
is in fact its right derivative) and 𝑉(𝑡)
≤ −𝐾(𝑉(𝑡))𝛼 , ∀𝑡 ≥ 0,
𝑉(𝑡0 ) ≥ 0, where 𝐾 > 0, 0 < 𝛼 < 1 are two constants. Then,
for any given 𝑡0 , 𝑉(𝑡) satisfies the following inequality:
𝑉1−𝛼 (𝑡) ≤ 𝑉1−𝛼 (𝑡0 ) − 𝐾 (1 − 𝛼) (𝑡 − 𝑡0 ) ,
𝑉 (𝑡) ≡ 0, ∀𝑡 > 𝑡∗ ,
with 𝑡∗ given by 𝑡∗ = 𝑡0 + 𝑉1−𝛼 (𝑡0 )/𝐾(1 − 𝛼).
𝑡0 ≤ 𝑡 ≤ 𝑡∗ ,
(9)
Journal of Applied Mathematics
3
Lemma 7 (Jensen’s Inequality). If 𝑎1 , 𝑎2 , . . . , 𝑎𝑛 are positive
numbers and 0 < 𝑟 < 𝑝, then
(1/𝑝)
𝑝
(∑𝑎𝑖 )
𝑖=1
𝑛
≤
(1/𝑟)
𝑟
(∑𝑎𝑖 ) .
𝑖=1
𝑀𝑖 𝐸𝑒 (𝑡) 𝑊𝑖 = 𝑀𝑖 𝐸𝑦𝑖 (𝑡) = 𝑀𝑖 𝐸𝑄𝑖 𝑄𝑖−1 𝑦𝑖 (𝑡)
𝑦𝑖1
𝐼𝑟 0
𝑦1
=(
) ( 2) = ( 𝑖 ) ,
0 0
0
𝑦𝑖
𝑛
(10)
𝑇
3. Main Results
In this section, we consider the finite-time synchronization
of the singular coupled network (1) under the appropriate
controllers. In order to control the states of all nodes to the
synchronization state 𝑠(𝑡) in finite time, we apply some simple
controllers 𝑢𝑖 (𝑡) ∈ R𝑛 , 𝑖 = 1, 2, . . . , 𝑁, to system (1). Then,
the controlled system can be written as
𝐸𝑥𝑖̇ (𝑡) = 𝐴𝑥𝑖 (𝑡) + 𝑓 (𝑥𝑖 (𝑡) , 𝑡) + 𝑐 ∑𝑏𝑖𝑗 Γ𝑥𝑗 (𝑡) + 𝑢𝑖 ,
𝑗=1
(11)
𝑖 = 1, 2, . . . , 𝑁.
Then, we have
𝐸𝑒𝑖̇ (𝑡) = 𝐴𝑒𝑖 (𝑡) + 𝑓 (𝑥𝑖 (𝑡) , 𝑡) − 𝑓 (𝑠 (𝑡) , 𝑡)
𝑁
(12)
+ 𝑐 ∑ 𝑏𝑖𝑗 Γ𝑒𝑗 (𝑡) + 𝑢𝑖 ,
𝑗=1
𝐸𝑦𝑖̇ (𝑡) = (𝐴 + 𝑐𝜆 𝑖 Γ) 𝑦𝑖 (𝑡) + 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 + V𝑖 ,
(13)
where V𝑖 = 𝑢𝑊𝑖 , 𝑢 = (𝑢1 , 𝑢2 , . . . , 𝑢𝑁).
With Assumption 5, it follows from the proof of Theorem 1 in [2] and Lemma 2.2 in [3] that the pair (𝐸, 𝐴 + 𝑐𝜆 𝑖 Γ)
is regular and impulse-free; that is, there exist nonsingular
matrices 𝑀𝑖 , 𝑄𝑖 ∈ R𝑛×𝑛 satisfying that
sign (𝑦𝑖𝑟1 (𝑡)) , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
0, . . . , 0) .
𝑛−𝑟
Remark 8. From (17), the controllers 𝑢𝑖 are dependent not
only on the coupled matrix 𝐵, but also on the singular matrix
𝐸. And from the shape of controllers, we only use the states
𝑦𝑖1 of slow subsystems (15) in controllers V𝑖 , but we do not
consider the states 𝑦𝑖2 of fast subsystems (16). It is very special.
It is interesting for our future research to design more general
controller which makes the singular hybrid coupled networks
synchronize in finite time.
Theorem 9. Suppose that Assumptions 3, 4, and 5 hold.
Under the controllers (17), the singular system (1) is synchronized in a finite time 𝑡∗ = 𝑡0 + (𝑉(1−𝛽)/2 (𝑡0 )/
𝑇
𝑇
𝑎𝑘𝑏−(1+𝛽)/2 (1 − 𝛽)), where 𝑉(𝑡0 ) = ∑𝑁
𝑖=1 𝑦𝑖 (𝑡0 )𝐸 𝑃𝑖 𝑦𝑖 (𝑡0 ) =
𝑁
𝑇 𝑇
∑𝑖=1 (𝑒(𝑡0 )𝑊𝑖 ) 𝐸 𝑃𝑖 (𝑒(𝑡0 )𝑊𝑖 ), 𝑒(𝑡0 ) is the initial condition of
𝑒(𝑡), and 𝑎 and 𝑏 are defined as (25).
𝑦𝑖1̇ (𝑡) = 𝐴 𝑖 𝑦𝑖1 (𝑡) + 𝑀𝑖1 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 + 𝑀𝑖1 V𝑖 ,
(15)
0 = 𝑦𝑖2 (𝑡) + 𝑀𝑖2 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 + 𝑀𝑖2 V𝑖 ,
(16)
𝑦𝑖2 (𝑡)
𝑁
𝑉 (𝑡) = ∑𝑦𝑖𝑇 (𝑡) 𝐸𝑇 𝑃𝑖 𝑦𝑖 (𝑡) .
(19)
𝑖=1
where 𝐴 𝑖 ∈ R𝑟×𝑟 , 𝑖 = 1, 2, . . . , 𝑁. So, system (13) is equivalent
to
∈ R , and
𝑘 > 0 is a tunable constant, and the real number 𝛽 satisfies
0 < 𝛽 < 1. So, we obtain 𝑢 = (V1 , V2 , . . . , V𝑁)𝑊−1 =
V(𝜉1 , 𝜉2 , . . . , 𝜉𝑁). That is, 𝑢𝑖 = V𝜉𝑖 .
(14)
𝑀𝑖 (𝐴 + 𝑐𝜆 𝑖 Γ) 𝑄𝑖 = diag {𝐴 𝑖 , 𝐼𝑛−𝑟 } ,
𝑟
sign (𝑀𝑖 𝐸𝑦𝑖 (𝑡)) = diag ( sign (𝑦𝑖11 (𝑡)) , sign (𝑦𝑖21 (𝑡)) , . . . ,
Proof. Consider the following Lyapunov function:
𝑀𝑖 𝐸𝑄𝑖 = diag {𝐼𝑟 , 0} ,
𝑦1 (𝑡)
= ( 𝑦𝑖2 (𝑡) ), 𝑦𝑖1 (𝑡)
where
𝑖
𝑀1
𝑀𝑖 = ( 𝑀𝑖2 ), 𝑀𝑖1 ∈ R𝑟×𝑛 , 𝑀𝑖2 ∈
𝑖
R𝑛×𝑟 , and 𝑄𝑖2 ∈ R𝑛×(𝑛−𝑟) .
󵄨 1 󵄨𝛽
󵄨 1 󵄨𝛽
󵄨󵄨
󵄨𝛽
0, . . . , 0) ,
󵄨󵄨𝑀𝑖 𝐸𝑦𝑖 (𝑡)󵄨󵄨󵄨 = (󵄨󵄨󵄨󵄨𝑦𝑖1 (𝑡)󵄨󵄨󵄨󵄨 , . . . , 󵄨󵄨󵄨󵄨𝑦𝑖𝑟 (𝑡)󵄨󵄨󵄨󵄨 , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑛−𝑟
(18)
𝑁
𝑄𝑖−1 𝑦𝑖 (𝑡)
where
The derivative of 𝑉(𝑡) along the trajectory of system (13) is
𝑉̇ (𝑡)
𝑁
= ∑ [𝑦𝑖𝑇 (𝑡) 𝑃𝑖𝑇 ((𝐴 + 𝑐𝜆 𝑖 Γ) 𝑦𝑖 (𝑡) + 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 + V𝑖 )
𝑖=1
𝑇
∈R
R(𝑛−𝑟)×𝑛 , 𝑄𝑖 = ( 𝑄𝑖1
𝑛−𝑟
𝑄𝑖2
+((𝐴 + 𝑐𝜆 𝑖 Γ) 𝑦𝑖 (𝑡) + 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 + V𝑖 ) 𝑃𝑖 𝑦𝑖 (𝑡) ]
. And
), 𝑄𝑖1 ∈
In order to achieve our aim, we design the following
controllers:
󵄨󵄨𝛽
󵄨
V𝑖 = −𝑘𝑀𝑖−1 sign (𝑀𝑖 𝐸𝑒 (𝑡) 𝑊𝑖 ) 󵄨󵄨󵄨𝑀𝑖 𝐸𝑒 (𝑡) 𝑊𝑖 󵄨󵄨 ,
(17)
𝑁
𝑇
= ∑ [ 𝑦𝑖𝑇 (𝑡) (𝑃𝑖𝑇 (𝐴 + 𝑐𝜆 𝑖 Γ) + (𝐴 + 𝑐𝜆 𝑖 Γ) 𝑃𝑖 ) 𝑦𝑖 (𝑡)
𝑖=1
+ 2𝑦𝑖𝑇 (𝑡) 𝑃𝑖𝑇 𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖
󵄨𝛽
󵄨
−2𝑘𝑦𝑖𝑇 (𝑡) 𝑃𝑖𝑇 𝑀𝑖−1 sign (𝑀𝑖 𝐸𝑒 (𝑡) 𝑊𝑖 ) 󵄨󵄨󵄨𝑀𝑖 𝐸𝑒 (𝑡) 𝑊𝑖 󵄨󵄨󵄨 ].
(20)
4
Journal of Applied Mathematics
By using Assumption 4, one can get the following inequality:
󵄩󵄩 𝑁
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩 󵄩󵄩
󵄩󵄩𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ [𝑓 (𝑥𝑘 (𝑡) , 𝑡) − 𝑓 (𝑠 (𝑡) , 𝑡)] 𝑊𝑖𝑘 󵄩󵄩󵄩
󵄩󵄩𝑘=1
󵄩󵄩
󵄩
󵄩
𝑁
Let
𝑎 ≜ min {𝜆 min (𝑃𝑖1𝑇 )} ,
𝑖
From (22), we get
󵄩
󵄩
≤ ∑ 𝐿 𝑘 󵄩󵄩󵄩𝑒𝑘 (𝑡)󵄩󵄩󵄩
𝑁
𝑁
󵄩2
󵄩2
󵄩
󵄩
𝑎∑󵄩󵄩󵄩󵄩𝑦𝑖1 (𝑡)󵄩󵄩󵄩󵄩 ≤ 𝑉 (𝑡) ≤ 𝑏∑󵄩󵄩󵄩󵄩𝑦𝑖1 (𝑡)󵄩󵄩󵄩󵄩 .
𝑘=1
𝑁
𝑁
𝑘=1
𝑘=1
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
= ∑ 𝐿 𝑘 󵄩󵄩󵄩𝑦 (𝑡) 𝜉𝑘 󵄩󵄩󵄩 ≤ ∑ 𝐿 𝑘 󵄩󵄩󵄩𝑦 (𝑡)󵄩󵄩󵄩 = 𝐿 󵄩󵄩󵄩𝑦 (𝑡)󵄩󵄩󵄩
𝑖=1
󵄩
󵄩
≤ 𝐿 ∑ 󵄩󵄩󵄩𝑦𝑘 (𝑡)󵄩󵄩󵄩 ,
By the use of (24)–(26) and Lemma 7, we can obtain that
𝑖=1
𝑘=1
(21)
𝑇
𝑖
R𝑟×(𝑛−𝑟) , 𝑃𝑖3 ∈ R(𝑛−𝑟)×𝑟 , and 𝑃𝑖4 ∈ R(𝑛−𝑟)×(𝑛−𝑟) . Using (7) and
(8) (see [2]), one can obtain that 𝑃𝑖1 = (𝑃𝑖1 )𝑇 > 0 and 𝑃𝑖2 = 0;
then,
𝑉 (𝑡) =
−
𝑁
∑𝑦𝑖𝑇 (𝑡) 𝐸𝑇 𝑃𝑖 𝑦𝑖
𝑖=1
2𝑘𝑦𝑖𝑇 (𝑡) 𝑃𝑖𝑇 𝑀𝑖−1
(𝑡) =
𝑁
∑(𝑦𝑖1
𝑖=1
𝑇
(𝑡)) 𝑃𝑖1 𝑦𝑖1
(𝑡) ,
(22)
󵄨
󵄨𝛽
= −2𝑘𝑦𝑖𝑇 (𝑡) 𝑃𝑖𝑇 𝑀𝑖−1 (sign (𝑦𝑖11 (𝑡)) 󵄨󵄨󵄨󵄨𝑦𝑖11 (𝑡)󵄨󵄨󵄨󵄨 , . . . ,
=
𝑃𝑖1
𝑇
−𝑇
−2𝑘𝑦𝑖 (𝑡) 𝑄𝑖 ( 3
𝑃𝑖
0
𝑃𝑖4
󵄨𝛽
󵄨
(𝑡)) 󵄨󵄨󵄨󵄨𝑦𝑖𝑟1 (𝑡)󵄨󵄨󵄨󵄨 , 0, . . . , 0)
𝑇
) 𝑀𝑖 𝑀𝑖−1
From Lemma 6, we have that the solutions 𝑦𝑖1 (𝑡) of system
(15) are globally asymptotically stable with respect to 𝑦𝑖1 (𝑡) =
0 in the finite time 𝑡∗ ; that is,
󵄨𝛽 󵄩󵄩
󵄨
×󵄨󵄨󵄨𝑀𝑖 𝐸𝑦𝑖 (𝑡)󵄨󵄨󵄨 󵄩󵄩󵄩
󵄩
.
󵄩󵄩
󵄩2
󵄩
󵄩
󵄩 󵄩 󵄩
𝑉̇ (𝑡) ≤ ∑ [−𝜂𝑖 󵄩󵄩󵄩𝑦𝑖 (𝑡)󵄩󵄩󵄩 + 2𝐿 󵄩󵄩󵄩𝑃𝑖 󵄩󵄩󵄩 ∑ 󵄩󵄩󵄩󵄩𝑦𝑗 (𝑡)󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝑦𝑖 (𝑡)󵄩󵄩󵄩
𝑖=1
𝑗=1
[
𝑁
󵄨1+𝛽
󵄨
≤ ∑ − 2𝑘𝜆 min (𝑃𝑖1𝑇 ) 󵄨󵄨󵄨󵄨𝑦𝑖1 (𝑡)󵄨󵄨󵄨󵄨 .
𝑖=1
(30)
𝑁
󵄩
󵄩 󵄩
󵄩
󵄩 󵄩 󵄩
󵄩
≤ 𝐿∑ 󵄩󵄩󵄩󵄩𝑄𝑗 󵄩󵄩󵄩󵄩 (󵄩󵄩󵄩󵄩𝑦𝑗1 (𝑡)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑦𝑗2 (𝑡)󵄩󵄩󵄩󵄩) + 𝑘 󵄩󵄩󵄩󵄩𝑀𝑖−1 󵄩󵄩󵄩󵄩
(23)
󵄨1+𝛽
(𝑡)󵄨󵄨󵄨󵄨 ]
]
(29)
𝑁
󵄩 󵄩󵄩
󵄩
≤ 𝐿∑ 󵄩󵄩󵄩󵄩𝑦𝑗 (𝑡)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩 − 𝑘𝑀𝑖−1 sign (𝑀𝑖 𝐸𝑦𝑖 (𝑡))
󵄩
𝑗=1
𝑗=1
󵄨
󵄨𝛽
× 󵄨󵄨󵄨󵄨𝑦𝑖1 (𝑡)󵄨󵄨󵄨󵄨 .
𝑁
󵄨
−2𝑘𝜆 min (𝑃𝑖1𝑇 ) 󵄨󵄨󵄨󵄨𝑦𝑖1
(28)
Similar to the proof of Lemma 2.2 in [3] and the proof of
Theorem 1 in [17], let 𝑀𝑖2 (𝑀𝑖2 )𝑇 = 𝐼𝑛−𝑟 , which implies that
‖ 𝑀𝑖2 ‖= 1. One has
Substituting (8), (21), and (23) into (20) and letting 𝜂1 = 0,
while 𝜂𝑖 ≥ 2𝐿(𝑁 − 1) ‖ 𝑃𝑖 ‖, 𝑖 = 2, . . . , 𝑁, one has
𝑁
for 𝑡 ≥ 𝑡∗ ,
󵄩 󵄩
󵄩󵄩 2 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩
󵄩󵄩𝑦𝑖 (𝑡)󵄩󵄩 ≤ 󵄩󵄩𝑀𝑖 󵄩󵄩 󵄩󵄩𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑀𝑖2 󵄩󵄩󵄩 󵄩󵄩󵄩󵄩V𝑖 󵄩󵄩󵄩󵄩 .
󵄩 󵄩
󵄩
󵄩 󵄩 󵄩
󵄨
󵄨𝛽
= −2𝑘𝑦𝑖1𝑇 (𝑡) 𝑃𝑖1𝑇 sign (𝑦𝑖1 (𝑡)) 󵄨󵄨󵄨󵄨𝑦𝑖1 (𝑡)󵄨󵄨󵄨󵄨
󵄨
≤ −2𝑘𝜆 min (𝑃𝑖1𝑇 ) 󵄨󵄨󵄨󵄨𝑦𝑖1 (𝑡)󵄨󵄨󵄨
󵄩󵄩 1 󵄩󵄩
󵄩󵄩𝑦𝑖 (𝑡)󵄩󵄩 = 0
󵄩
󵄩
󵄩
󵄩
lim∗ 󵄩󵄩󵄩󵄩𝑦𝑖1 (𝑡)󵄩󵄩󵄩󵄩 = 0,
𝑡→𝑡
󵄩󵄩 2 󵄩󵄩 󵄩󵄩
󵄩󵄩𝑦𝑖 (𝑡)󵄩󵄩 ≤ 󵄩󵄩𝐹 (𝑒 (𝑡) , 𝑡) 𝑊𝑖 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑢𝑖 𝑊𝑖 󵄩󵄩󵄩󵄩
󵄩
󵄩
󵄨󵄨 1 󵄨󵄨𝛽
󵄨
󵄨
× diag (sign (𝑦𝑖1 (𝑡)) , 0) (󵄨󵄨𝑦𝑖 (𝑡)󵄨󵄨 )
0
󵄨󵄨1+𝛽
𝑖=1
where 𝑡∗ = 𝑡0 + (𝑉(1−𝛽)/2 (𝑡0 )/𝑎𝑘𝑏−(1+𝛽)/2 (1 − 𝛽)).
In the following, we show that 𝑦𝑖2 (𝑡) are globally asymptotically stable with respect to 𝑦𝑖2 (𝑡) = 0 in the finite time 𝑡∗ .
From (16), one has
󵄨𝛽
󵄨
sign (𝑀𝑖 𝐸𝑒 (𝑡) 𝑊𝑖 ) 󵄨󵄨󵄨𝑀𝑖 𝐸𝑒 (𝑡) 𝑊𝑖 󵄨󵄨󵄨
sign (𝑦𝑖𝑟1
(1+𝛽)/2
(1+𝛽)/2
1
≤ − 2𝑎𝑘( 𝑉 (𝑡))
= −2𝑎𝑘𝑏−(1+𝛽)/2 (𝑉 (𝑡))(1+𝛽)/2 .
𝑏
(27)
−1
where 𝑊𝑖 = (𝑊𝑖1 , 𝑊𝑖2 , . . . , 𝑊𝑖𝑛 ) and (𝜉1 , 𝜉2 , . . . , 𝜉𝑁) = 𝑊 =
𝑊𝑇 .
𝑃1 𝑃2
Define 𝑀𝑖−𝑇 𝑃𝑖 𝑄𝑖 = ( 𝑃𝑖3 𝑃𝑖4 ), where 𝑃𝑖1 ∈ R𝑟×𝑟 , 𝑃𝑖2 ∈
(26)
𝑖=1
𝑁
𝑁
󵄨1+𝛽
󵄨
󵄩2
󵄩
𝑉̇ (𝑡) ≤ − 2𝑎𝑘∑󵄨󵄨󵄨󵄨𝑦𝑖1 (𝑡)󵄨󵄨󵄨󵄨
≤ −2𝑎𝑘(∑󵄩󵄩󵄩𝑦𝑖 (𝑡)󵄩󵄩󵄩 )
𝑁
𝑖
𝑏 ≜ max {𝜆 max (𝑃𝑖1𝑇 )} . (25)
𝑖
Then,
(24)
𝑁
𝑁
󵄩 𝑁
󵄩 󵄩
󵄩
󵄩
󵄩 󵄩 󵄩
∑ 󵄩󵄩󵄩󵄩𝑦𝑖2 (𝑡)󵄩󵄩󵄩󵄩 ≤ ∑ [𝐿∑ 󵄩󵄩󵄩󵄩𝑄𝑗 󵄩󵄩󵄩󵄩 (󵄩󵄩󵄩󵄩𝑦𝑗1 (𝑡)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑦𝑗2 (𝑡)󵄩󵄩󵄩󵄩)
𝑖=1
𝑖=1
[ 𝑗=1
󵄩
󵄨𝛽
󵄩󵄨
+𝑘 󵄩󵄩󵄩󵄩𝑀𝑖−1 󵄩󵄩󵄩󵄩 󵄨󵄨󵄨󵄨𝑦𝑖1 (𝑡)󵄨󵄨󵄨󵄨 ] ;
]
(31)
Journal of Applied Mathematics
5
that is,
𝑁
󵄩 𝑁
󵄩
󵄩 󵄩 󵄩
󵄩 󵄩󵄩
∑ (1 − 𝑁𝐿 󵄩󵄩󵄩𝑄𝑖 󵄩󵄩󵄩) 󵄩󵄩󵄩󵄩𝑦𝑖2 (𝑡)󵄩󵄩󵄩󵄩 ≤ ∑ (𝑁𝐿 󵄩󵄩󵄩𝑄𝑖 󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑦𝑗1 (𝑡)󵄩󵄩󵄩󵄩
𝑖=1
𝑖=1
(32)
󵄩
󵄨𝛽
󵄩󵄨
+𝑘 󵄩󵄩󵄩󵄩𝑀𝑖−1 󵄩󵄩󵄩󵄩 󵄨󵄨󵄨󵄨𝑦𝑖1 (𝑡)󵄨󵄨󵄨󵄨 ) .
With Assumption 4, there must exist nonsingular matrices 𝑀𝑖 , 𝑄𝑖 ∈ R𝑛×𝑛 satisfying the equalities 𝑀𝑖 𝐸𝑄𝑖 =
diag{𝐼𝑟 , 0}, 𝑀𝑖 (𝐴+𝑐𝜆 𝑖 Γ)𝑄𝑖 = diag{𝐴 𝑖 , 𝐼𝑛−𝑟 }, where 𝐴 𝑖 ∈ R𝑟×𝑟 ,
𝑖 = 1, 2, . . . , 𝑁. Moreover, nonsingular matrices 𝑄𝑖 can be
suitably chosen to satisfy 1 − 𝑁𝐿 ‖ 𝑄𝑖 ‖> 0, for ∀𝑖 ∈
{1, 2, . . . , 𝑁}. Therefore, one can obtain lim𝑡 → 𝑡∗ ‖ 𝑦𝑖2 (𝑡) ‖= 0,
and ‖ 𝑦𝑖2 (𝑡) ‖= 0 for 𝑡 ≥ 𝑡∗ from (28) and (32), 𝑖 = 1, 2, . . . , 𝑁.
Consequently, lim𝑡 → 𝑡∗ ‖ 𝑒𝑖 (𝑡) ‖= 0, and ‖ 𝑒𝑖 (𝑡) ‖= 0 for
𝑡 ≥ 𝑡∗ , 𝑖 = 1, 2, . . . , 𝑁. The proof is completed.
If rank(𝐸) = 𝑛, system (1) is a general nonsingular
coupled network. By using the controllers 𝑢𝑖 similar to V𝑖 in
(17), we can derive the finite-time synchronization of system
(1). For simplicity, let 𝐸 = 𝐼𝑛 . Then, we have the following.
Corollary 10. When 𝐸 = 𝐼𝑛 , under Assumptions 3 and 4, let
the controllers 𝑢𝑖 (𝑡) be as follows:
󵄨
󵄨𝛽
𝑢𝑖 (𝑡) = −𝑘 sign (𝑒𝑖 (𝑡)) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 ,
𝑖 = 1, 2, . . . , 𝑁;
(33)
system (1) is synchronized in a finite time.
Remark 11. Since the conditions in Assumption 5 are not
strict LMIs problems, they cannot be solved directly by
the LMI Matlab Toolbox. According to Lemma 1 in [17],
Lemma 1 in [9], and Remark 3 in [18], if matrix 𝐸 has the
decomposition as
Suppose that we choose the average state of all node states
as synchronized state; that is, 𝑠(𝑡) = (1/𝑁) ∑𝑁
𝑘=1 𝑥𝑘 (𝑡). We
have similar results. Before giving these results, we need some
assumptions as follows:
Assumption 2󸀠 . Assume that there exist nonnegative constants 𝐿 𝑖𝑗 such that
󵄩
󵄩
󵄩󵄩
󵄩
󵄩󵄩𝑓 (𝑥𝑖 (𝑡) , 𝑡) − 𝑓 (𝑥𝑗 (𝑡) , 𝑡)󵄩󵄩󵄩 ≤ 𝐿 𝑖𝑗 󵄩󵄩󵄩𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡)󵄩󵄩󵄩 ,
󵄩
󵄩
󵄩
󵄩
(36)
𝑖, 𝑗 = 1, 2, . . . , 𝑁.
Assumption 3󸀠 . There exist matrices 𝑃𝑖 such that
𝐸𝑇 𝑃𝑖 = 𝑃𝑖𝑇 𝐸 ≥ 0,
𝑖 = 1, 2, . . . , 𝑁,
𝐴𝑇 𝑃1 + 𝑃1𝑇 𝐴 < 0,
𝑇
(𝐴 + 𝑐𝜆 𝑖 Γ) 𝑃𝑖 + 𝑃𝑖𝑇 (𝐴 + 𝑐𝜆 𝑖 Γ) ≤ −𝜍𝑖 𝐼,
𝑖 = 2, . . . , 𝑁,
where 𝜍𝑖 > (4𝐿/𝑁)(𝑁 − 1) ‖ 𝑃𝑖 ‖, 𝐿 = ∑𝑁
𝑖=1 𝐿 𝑖 , and 𝐿 𝑖 =
𝑁
∑𝑘=1,𝑘 ≠ 𝑖 𝐿 𝑖𝑘 .
Theorem 13. Suppose that Assumptions 3, 2󸀠 , and 3󸀠 hold. By
the controllers (17), the singular hybrid coupled network (1) can
be synchronized to the average state of all node states in the
finite time in the sense of Definition 2.
Corollary 14. Suppose that Assumptions 3 and 2󸀠 hold, and
matrix 𝐸 has the decomposition as (34) in Remark 11. By the
controllers (17), if there exist matrices 𝑇𝑖 ∈ R𝑟×𝑟 , 𝑇𝑖 ≥ 0, and
𝑆𝑖 ∈ R(𝑛−𝑟)×𝑟 , 𝑖 = 1, 2, . . . , 𝑁, such that
𝑇
𝐴𝑇 (𝑈1 𝑇1 𝑈1𝑇 𝐸 + 𝑈2 𝑆1 ) + (𝑈1 𝑇1 𝑈1𝑇𝐸 + 𝑈2 𝑆1 ) 𝐴 < 0,
𝑇
𝐼 0 Ξ 0
) 𝑉𝑇 ,
𝐸 = 𝑈( 𝑟 )( 𝑟
0 0
0 𝐼𝑛−𝑟
(34)
where 𝑈 = (𝑈1 , 𝑈2 ), 𝑉 = (𝑉1 , 𝑉2 ), and Ξ𝑟 = diag{󰜚1 , . . . , 󰜚𝑟 }
with 󰜚𝑖 > 0 for 𝑖 = 1, 2, . . . , 𝑟, then Assumption 5 can be
transformed into a strict LMIs problem.
Corollary 12. Suppose that Assumptions 3 and 4 hold, and
matrix 𝐸 has the decomposition as (34) in Remark 11. By the
controllers (17), the singular hybrid coupled network (1) can be
synchronized in the finite time in the sense of Definition 2, if
there exist matrices 𝑇𝑖 ∈ R𝑟×𝑟 , 𝑇𝑖 ≥ 0, and 𝑆𝑖 ∈ R(𝑛−𝑟)×𝑟 ,
𝑖 = 1, 2, . . . , 𝑁, such that
𝑇
𝐴𝑇 (𝑈1 𝑇1 𝑈1𝑇 𝐸 + 𝑈2 𝑆1 ) + (𝑈1 𝑇1 𝑈1𝑇 𝐸 + 𝑈2 𝑆1 ) 𝐴 < 0,
𝑇
(𝐴 + 𝑐𝜆 𝑖 Γ) (𝑈1 𝑇𝑖 𝑈1𝑇 𝐸 + 𝑈2 𝑆𝑖 )
(35)
(37)
(𝐴 + 𝑐𝜆 𝑖 Γ) (𝑈1 𝑇𝑖 𝑈1𝑇 𝐸 + 𝑈2 𝑆𝑖 )
(38)
𝑇
+ (𝑈1 𝑇𝑖 𝑈1𝑇 𝐸 + 𝑈2 𝑆𝑖 ) (𝐴 + 𝑐𝜆 𝑖 Γ) ≤ −𝜂𝑖 𝐼,
where 𝜂𝑖 > (4𝐿/𝑁)(𝑁 − 1) ‖ 𝑃𝑖 ‖, 𝑃𝑖 = (𝑈1 𝑇𝑖 𝑈1𝑇 𝐸 + 𝑈2 𝑆𝑖 ),
𝑁
𝑖 = 2, . . . , 𝑁, 𝐿 = ∑𝑁
𝑖=1 𝐿 𝑖 , and 𝐿 𝑖 = ∑𝑘=1,𝑘 ≠ 𝑖 𝐿 𝑖𝑘 , the singular
hybrid coupled network (1) can be synchronized to the average
state of all node states in the finite time.
Remark 15. In this paper, we study finite-time synchronization of the singular hybrid coupled networks when the singular systems studied in this paper are assumed to be regular
and impulse-free. However, it may be more complicated when
we do not assume in advance that the systems are regular and
impulse free. Synchronization or finite-time synchronization
of singular coupled systems is worth discussing without
the assumption that the considered systems are regular and
impulsive free.
𝑇
+ (𝑈1 𝑇𝑖 𝑈1𝑇 𝐸 + 𝑈2 𝑆𝑖 ) (𝐴 + 𝑐𝜆 𝑖 Γ) ≤ −𝜂𝑖 𝐼,
where 𝜂𝑖 > 2𝐿(𝑁 − 1) ‖ 𝑃𝑖 ‖, 𝑃𝑖 =
2, . . . , 𝑁, and 𝐿 = ∑𝑁
𝑖=1 𝐿 𝑖 .
(𝑈1 𝑇𝑖 𝑈1𝑇 𝐸
+ 𝑈2 𝑆𝑖 ), 𝑖 =
4. An Illustrative Example
In this section, a numerical example will be given to verify the
theoretical results obtained earlier.
6
Journal of Applied Mathematics
2.5
Example 16. Consider the following singular hybrid coupled
network which is similar to one given in [18]:
2
6
𝐸𝑥𝑖̇ (𝑡) = 𝐴𝑥𝑖 (𝑡) + 𝑓 (𝑥𝑖 (𝑡) , 𝑡) + 𝑐 ∑ 𝑏𝑖𝑗 Γ𝑥𝑗 (𝑡) + 𝑢𝑖 ,
𝑗=1
1.5
(39)
1
where 𝑥𝑖 (𝑡) = (𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡))𝑇 , 𝑓(𝑥𝑖 (𝑡), 𝑡) = ((1/
15) tanh(𝑥𝑖1 (𝑡)), tanh(𝑥𝑖2 (𝑡)))𝑇 , 𝑠(𝑡) = (0, 0)𝑇 , 𝐿 𝑖 = 1/15,
𝐿 = 2/5, 𝑐 = 1, and
8 0
𝐸=(
),
0 0
−10 1
𝐴=(
),
1 −10
−5
1
1
𝐵=(
1
1
(1
1
−4
1
1
1
0
1
1
−4
1
0
1
1
1
1
−4
1
0
1
1
0
1
−4
1
𝑒𝑖1 (𝑡)
𝑖 = 1, 2, . . . , 6,
0
−0.5
1 0
Γ=(
),
0 1
1
0
1)
.
0
1
−3)
0.5
−1
−1.5
0
2
4
6
8
10
𝑡
Figure 1: Error variable 𝑒𝑖1 (𝑡) (𝑖 = 1, 2, . . . , 6) of system (39) with
𝑘 = 1.
(40)
Since 𝐵 is symmetric matrix and its six eigenvalues are 𝜆 1 =
0, 𝜆 2 = −3, 𝜆 3 = −4, 𝜆 4 = −5, 𝜆 5 = −6,and 𝜆 6 = −6,
there exists a unitary matrix
0.25
0.2
𝑊 = (𝑊1 , . . . , 𝑊6 )
such that 𝐵 = 𝑊Λ𝑊𝑇 and Λ = diag(0, −3, −4, −5, −6, −6).
Choose
1
0
1
8
1
𝑀𝑖 = ( 10 − 𝜆 𝑖 ) ,
),
𝑄𝑖 = (
1
1
0
1
8 (10 − 𝜆 𝑖 ) 𝜆 𝑖 − 10
1 0
𝑃𝑖 = (
),
0 1
(42)
𝜂𝑖 > 4, 𝑖 = 1, 2, . . . , 6, and 𝛽 = 1/2 satisfying Assumptions
3, 4, and 5. Under the controllers 𝑢𝑖 defined in Theorem 9,
the singular hybrid system (39) can be synchronized in the
finite time 𝑡∗ = 7.1858 according to Theorem 9 if 𝑘 = 1. If the
controller gain 𝑘 = 5, 𝑡∗ = 1.4372. Corresponding simulation
results are shown in Figures 1 and 2 with initial conditions
1 3 2 0.7 2.5 0.3 ).
𝑒(0) = ( 0.5
−1 0.8 −2 1.5 −0.5
0.15
0.1
𝑒𝑖2 (𝑡)
1
2
1
0
0
0 −
−
−
√6
√6 √6
1
1
1
1
(−
0 −
0 )
)
( √
√
√
√
6
6
2
)
( 6
)
(
1 )
1
1
( 1
) (41)
(−
0 −
0 −
( √6
√2
√6 √6 )
)
(
=(
)
1
1
1
(− 1
0
0 )
)
( √
√2 √6
)
( 6 √6
)
(
1 )
1
1
( 1
)
(−
0 −
0
√6
√2
√6 √6
1
2
1
−
−
0
0
0
√
√
√
)
( 6
6
6
0.05
0
−0.05
−1
0
2
4
6
8
10
𝑡
Figure 2: Error variable 𝑒𝑖2 (𝑡) (𝑖 = 1, 2, . . . , 6) of system (39) with
𝑘 = 1.
5. Conclusions
In this paper, we discuss finite-time synchronization of the
singular hybrid coupled networks with the assumption that
the considered singular systems are regular and impulsivefree. Some sufficient conditions are derived to ensure finitetime synchronization of the singular hybrid coupled networks under a state feedback controller by finite-time stability
theory. A numerical example is finally exploited to show the
effectiveness of the obtained results. It will be an interesting
topic for the future researches to extend new methods to
study synchronization, robust control, pinning control, and
finite-time synchronization of singular hybrid coupled networks without the assumption that the considered singular
systems are regular and impulsive-free.
Journal of Applied Mathematics
Acknowledgments
This work was jointly supported by the National Natural
Science Foundation of China under Grant nos. 61272530
and 11072059 and the Jiangsu Provincial Natural Science
Foundation of China under Grant no. BK2012741.
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